Does F(x)=9-x^2 satisfy the hypothesis of the mean value theorem on the interval [0,3]? if it does, then find the exact values of all Ce(0,3) that satisfy the conclusion of the Mean Value Theorem.
f(3) = 0
f(0) = 9
slope = (0-9)/3 = -3
f' = -2x
where is -2x = -3 ?
x = 1.5
To determine if the function f(x) = 9 - x^2 satisfies the hypothesis of the mean value theorem on the interval [0,3], we need to check two conditions:
1. Continuity: The function f(x) must be continuous on the closed interval [0,3].
2. Differentiability: The function f(x) must be differentiable on the open interval (0,3).
Let's check both conditions:
1. Continuity:
To determine if f(x) is continuous on [0,3], we need to check if it is continuous at the endpoints (0 and 3) and within the interval.
- At the endpoint x = 0:
lim(x->0+) f(x) = lim(x->0+) (9 - x^2) = 9
f(0) = 9
- At the endpoint x = 3:
lim(x->3-) f(x) = lim(x->3-) (9 - x^2) = 0
f(3) = 9 - 3^2 = 0
Since both the function and its limit at the endpoints are equal, f(x) is continuous throughout the interval [0,3].
2. Differentiability:
To determine if f(x) is differentiable on (0,3), we need to check if it is differentiable within the interval.
The derivative of f(x) = 9 - x^2 is:
f'(x) = -2x
Since the derivative -2x is defined and continuous on the open interval (0,3), f(x) is differentiable within the interval.
Therefore, f(x) = 9 - x^2 satisfies the hypothesis of the mean value theorem on the interval [0,3].
To find the exact values of all c in (0,3) that satisfy the conclusion of the mean value theorem, we need to find the average rate of change of the function over the interval [0,3] and find a c value where the instantaneous rate of change (derivative) matches the average rate of change.
The average rate of change (slope) over the interval [0,3] is given by the difference quotient:
Average rate of change = (f(3) - f(0))/(3 - 0)
Substituting the function values:
Average rate of change = (0 - 9)/(3 - 0) = -3
To find a c value, we need to find where f'(c) = -3. We know that f'(x) = -2x, so:
-2c = -3
Solving for c:
c = 3/2
So, the exact values of c from (0,3) that satisfy the conclusion of the Mean Value Theorem are c = 3/2 or 1.5.